We write \(S_n\)  for the symmetric group on \(n\)  letters.
For background on the representation theory of the symmetric groups, the reader is referred to [1]}, [2]}.
For \(\lambda \in P(n)\) , we let \(S^\lambda _\mathbb {Z}\)  and \(M^\lambda _\mathbb {Z}\)  denote the Specht and Young permutation modules for \(S_n\) 
associated to \(\lambda \) , defined over \(\mathbb {Z}\) .
Given \(F\)  a field, we set \(S^\lambda _F=S^\lambda _\mathbb {Z}\otimes _\mathbb {Z}F \) 
and \(M^\lambda _F=M^\lambda _\mathbb {Z}\otimes _\mathbb {Z}F \) .
The permutation module \(M^\lambda _\mathbb {Z}\)  can be constructed as the set of \({\sf RStd}(\lambda )\)  under the symmetric group action by place permutation (modulo the equivalence class \(\sim \) ).
By classical rules of Young and Littlewood–Richardson, we have that
\(M^\lambda _{\mathbb {C}} \cong \bigoplus _{\nu \in P(n)} |{\sf SStd}( \lambda ,\nu )| S^\nu _{\mathbb {C}} \qquad {\rm ind}_{S_m\times S_n}^{S_{m+n}}(S^\lambda _{\mathbb {C}} \boxtimes S^\mu _{\mathbb {C}}) \cong \!\!\!\bigoplus _{\nu \in P(m+n)} |{\sf LR}(\nu \setminus \lambda ,\mu )| S^\nu _{\mathbb {C}}.\) 
